Impact of nano-size weathering products on the dissolution rates of primary minerals

Chemical Geology 282 (2011) 11–18

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Chemical Geology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c h e m g e o

Impact of nano-size weathering products on the dissolution rates of primary minerals Simon Emmanuel a,⁎, Jay J. Ague b a b

Institute of Earth Sciences, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem, 91904 Israel Department of Geology and Geophysics, Yale University, P.O. Box 208109, New Haven, CT 06520-8109 USA

a r t i c l e

i n f o

Article history: Received 7 June 2010 Received in revised form 1 January 2011 Accepted 3 January 2011 Available online 10 January 2011 Editor: J. Fein Keywords: Reaction kinetics Interfacial free energy Crystallization Precipitation

a b s t r a c t The natural weathering rates of primary minerals are often orders of magnitude lower than the rates of mineral dissolution measured in laboratory experiments. Primary dissolution rates are thought to be determined by the rate of secondary mineral precipitation, and in this paper we present a new approach to quantify the role played by interfacial energy, crystal size, and degree of supersaturation on precipitation kinetics in a population of crystals growing in a supersaturated fluid. We demonstrate that net mineral precipitation rates in systems that are close to equilibrium, and which possess a large number of micron and nanometer scale crystals, can be much lower than the rates predicted by standard kinetic equations. Moreover, when crystals are small enough, net dissolution dominates even when the system is supersaturated with respect to large crystals so that the standard reaction rate models used to describe bulk rates will no longer apply. Importantly, secondary minerals that form from the incongruent dissolution of primary phases are often submicron in size and field conditions are often far closer to equilibrium than those typically encountered in laboratory experiments. Thus, we propose that standard kinetic models – which ignore interfacial energy effects in small crystals – may be unsuitable to describe reaction kinetics in weathering systems. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The rates of mineral weathering derived from field-based studies are often orders of magnitude lower than the rates of mineral dissolution measured in laboratory experiments (Schnoor, 1990; van Grinsven and van Riemsdijk, 1992; Anbeek, 1993; Casey et al., 1993; Velbel, 1993; Blum and Stillings, 1995; Baxter and DePaolo, 2000; Oelkers et al., 2000; Yokoyama and Banfield, 2002; Baxter, 2003; Yoo and Mudd, 2008). Such apparent inconsistencies are usually attributed to differences in fluid chemistry, as well as additional mechanisms, including the depletion of reactive surfaces during weathering, the accumulation of surface coatings on primary minerals, and the slow rate of secondary mineral formation (e.g., White and Brantley, 2003; Maher et al., 2006, 2009; Zhu and Lu, 2009). While such mechanisms are likely to contribute to lower reaction rates in many natural weathered systems, a complete understanding of the relationship between field observations and experimental measurements has yet to be achieved. The weathering of primary minerals often involves incongruent dissolution during which the primary mineral dissolves and a new secondary mineral phase forms. Critically for reaction kinetics, it has

⁎ Corresponding author. E-mail address: [email protected] (S. Emmanuel). 0009-2541/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemgeo.2011.01.002

been demonstrated both theoretically and empirically that the weathering rate of a primary mineral is dependent on the rate of formation of the secondary phase (Alekseyev et al., 1997; Lasaga, 1998; Ganor et al., 2007; Zhu and Lu, 2009). As a result, if the precipitation rate is low, the dissolution rate of the primary mineral will also be limited. Such a state, however, will only occur once steady state has been reached, which is often not the case in typical laboratory experiments, and it has been proposed that this could account for the difference between lab and field rates. While this and other mechanisms are likely to limit weathering rates in field settings, here we explore the possibility that additional factors could be at play. In standard expressions used to describe bulk reaction rates, the rate of mineral precipitation is often governed by the reaction rate coefficient, reactive mineral surface area, and the degree of supersaturation. However, such standard rate equations may not always be appropriate in natural systems, particularly in systems in which nanometer and micron size crystals are common. Due to interfacial energy effects, small crystals are typically far more soluble than large crystals, a phenomenon responsible for the Ostwald ripening process during which large crystals grow at the expense of small ones (e.g., Steefel and Van Cappellen, 1990). Recently it has been shown that interfacial energy effects in rocks with a significant proportion of nanoscale and micron size pores can strongly reduce the reactive surface area, thereby lowering the bulk rate of mineral precipitation (Emmanuel et al., 2010). Micron and nanometer size weathering

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products, especially clay minerals and iron and aluminium oxides, are ubiquitous in soil profiles (e.g., Jepson and Rowse, 1975; Dorn, 1995; Melo et al., 2001; Wiriyakitnateekul et al., 2007), and in this paper we explore the way in which interfacial energy effects associated with these tiny phases could control the rate of secondary phase precipitation and the overall rate of weathering. 2. Theory 2.1. Dependence of weathering rates on secondary mineral precipitation To illustrate the effect of the size of weathering products on the dissolution of primary minerals, it is instructive to first consider the weathering of a mineral, such as feldspar, which may dissolve incongruently to form kaolinite and subsequently gibbsite. This process can be represented by two separate reactions: þ

þ

KAlSi3 O8 ðsÞ + 2H ðaqÞ + 9H2 OðlÞ⇒2K ðaqÞ + 4H4 SiO4 ðaqÞ

ð1Þ

feldspar

+ Al2 Si2 O5 ðOHÞ4 ðsÞ ; kaolinite

that the choice of model will alter the specifics of the calculations but not the conclusions of the present study. In this standard formulation, the crystal size distribution can affect the rate equation by influencing the amount of available surface area on which the mineral can precipitate (Steefel and Van Cappellen, 1990). However, in systems in which interfacial energy effects are significant, a fundamentally different kinetic approach is required. It has long been recognized that solubility is strongly affected by crystal size, with crystals dissolving more readily as their size decreases. This effect is related to the change in interfacial energy of the growing crystal and for a crystal of characteristic size r, the effective solubility, Sr, can be given by (Adamson, 1990; Scherer, 2004)

Sr = S0 exp

kaolinite

ð2Þ

Sr = S0 exp

gibbsite

Upon exposure to a primary mineral, the concentrations of dissolved species initially increase in an undersaturated fluid phase. When a relatively high level of supersaturation is achieved with respect to the weathering products, nucleation and growth of secondary mineral crystals occur. As a result the concentrations of dissolved species fall, and, all else being equal, dissolved concentrations should continue to drop until steady state is achieved (Fritz et al., 2009). When feldspar weathers irreversibly, it can be shown that at steady-state the rate of secondary mineral precipitation is linearly dependent on the rate of dissolution, such that the rate of feldspar dissolution, Rfeldspar, is theoretically equal to Rfeldspar = Rgibbsite + 2Rkaolinite, where Rgibbsite and Rkaolinite are the rates of precipitation of gibbsite and kaolinite respectively (Lasaga, 1998). This example serves to demonstrate the dependence of dissolution rates on secondary mineral formation, and other studies have also concluded that the rate of secondary mineral precipitation represents the rate determining step in primary mineral weathering (Alekseyev et al., 1997; Ganor et al., 2007; Zhu and Lu, 2009).

ð5Þ

where νm is the molar volume of the mineral, Rg is the gas constant, T is temperature, and γ is the interfacial energy. Importantly, ζ represents the crystal curvature, defined as the rate of change of area with respect to volume; thus, for a spherical crystal possessing a radius of r, ζ = 2/r and Eq. (5) becomes

and Al2 Si2 O5 ðOHÞ4 ðsÞ + 5H2 OðlÞ⇌ 2AlðOHÞ3 ðsÞ + 2H4 SiO4 ðaqÞ:

! νm γζ ; Rg T

! 2ν m γ : Rg Tr

ð6Þ

Crucially, Eq. (6) predicts that smaller crystals will be more soluble than large ones, although it can be shown that this effect only becomes significant in micron and nanometer size crystals. Furthermore, although the equation is developed for spherical crystals, such size effects can be demonstrated for all common crystal shapes. While Eq. (6) was developed for homogeneous growth of spherical crystals, a similar equation can be developed for the heterogenous growth of semispherical crystals on a planar substrate. In this case, γ can be replaced by an effective interfacial energy, γe, defined by γe = γcf + (γcs − γsf)/4, where the subscripts cf, cs, and sf indicate crystal–fluid, crystal–substrate, and substrate–fluid interfaces respectively. Thus, when γcs = γsf, the heterogeneous and homogenous growth models are equivalent. For systems containing an existing distribution of crystal sizes, the size dependence of solubility due to interfacial energy effects means that Eq. (4) may no longer represent the bulk precipitation rate; instead, the overall rate is given as the sum of the growth rates associated with each crystal size. Assuming a continuous distribution of crystal sizes, the effective rate, Re, is given by

2.2. Kinetic theory and interfacial energy effects ∞

The rate of precipitation, R, of a mineral phase in a system close to saturation is often given by a rate law of the form (Lasaga, 1998) R = k A f ðΔGÞ:

ð3Þ

Here, k is a rate coefficient, A is the specific surface area of the precipitating mineral, and f(ΔG) is some function of the free energy of reaction. This relationship is often defined as f(ΔG) = (S/S0 − 1)β, where S is the ion activity product, S0 is the bulk solubility product, and β is the reaction rate order. Thus, for many reactions a generic rate equation can be defined by β

R = k A ðS= S0 −1Þ :

ð4Þ

While this model adequately describes the behavior of some minerals, such as quartz, more complex functional forms for f(ΔG) have been proposed for other phases (e.g., Oelkers et al., 1994; Schott et al., 2009). However, all these models involve some function of S/S0, so

β

Re = k∫ Ar ðS=Sr −1Þ dr; 0

ð7Þ

where Ar is the probability density function of surface area expressed in terms of crystal size r. Thus, the rate expression is transformed from an ordinary differential equation into an integro-differential equation, and substitution of the size dependent relationship for the solubility of spherical crystals yields ∞

Re = k∫ Ar 0



β   S exp −2ν m γ =Rg Tr −1 dr: S0

ð8Þ

We emphasize that although Eq. (8) is an integro-differential equation, when interfacial energy effects can be neglected – as will occur when crystals are large or when the system is far from equilibrium – it will simplify to the ordinary differential equation given in Eq. (4). To help generalize the approach, we identify two parameters that control the magnitude of the interfacial energy effect on reaction rates:

S. Emmanuel, J.J. Ague / Chemical Geology 282 (2011) 11–18

εr [log10 m]

0 μ = −11.5 μ = −13.8

−5

μ = −16.1 −10 10

−8

10

−6

10

−4

r [m]

b

0.01 μ = −11.5

∫r0 εr dr

The Λ parameter – similar to that described by Emmanuel et al. (2010) – has dimensions of length and can be defined for many mineral systems; in Table 1, the values of Λ for common secondary phases are given. Although this kinetic expression remains untested under laboratory conditions, Emmanuel et al. (2010) found that a similar expression could account for the lack of quartz precipitation in the micron scale pores of a mineralized sandstone; in contrast, standard kinetic formulations failed to reproduce the observed patterns. Thus, the definition of the bulk reaction rate presented in this study allows for complex system behavior that is not predicted by standard kinetic approaches, and this aspect of the model is explored in the next section. Presently, one of the main limitations of the model is that it does not fully reflect the complexity of crystal growth kinetics on actual surfaces. Crystal surfaces are often highly heterogeneous, being composed of kink sites, defects, and steps, and incorporating such features into kinetic models remains a non-trivial challenge. However, as such features will presumably lead to local variations in interfacial energy, it is possible that they might be incorporated into the current model via the surface energy term. At present, further study is required to characterize the energetic effects of surface features in micron and nanometer scale crystals. Our approach, summarized in Fig. 2, is similar in some respects to the recently developed NANOKIN code, which has been used to simulate the kinetics of nucleation and crystal population growth in geochemical systems (Fritz et al., 2009). However, rather than simulate system evolution, the aim of this paper is to demonstrate the way in which interfacial energy effects – which are ultimately governed by crystal size and the level of fluid supersaturation – can influence bulk weathering rates.

μ = −13.8 0.005

μ = −16.1

0 10

−8

10

−6

10

−4

r [m]

c

6

−1

ð9Þ

10

2ν m γ : Rg T

5

4 b μ = −11.5

2

μ = −13.8 0

μ = −16.1

r

Λ=

a

∫0 A dr [log m ]

(i) bulk supersaturation (S/S0), and (ii) an interfacial energy parameter, Λ, defined here as

13

−2 10

−8

10

−6

10

−4

r [m] Fig. 1. (a) Probability density function for crystal volume fraction, r for different lognormal distributions. (b) Cumulative crystal volume fraction as a function of crystal radius for different distributions. (c) Cumulative surface area as a function of crystal size for different distributions. Values of μ = − 11.5, –13.8, and − 16.1 correspond to mean radii of 10− 5 m, 10− 6 m, 10− 7 m respectively.

2.3. Quantifying the effect of interfacial energy on precipitation rates In natural systems, crystal sizes can vary by orders of magnitude, with size distributions often exhibiting lognormal characteristics (e.g., Eberl et al., 1998, 2002). In such cases, the continuous density function for crystal sizes, r, is given by r =

!  −ðln r−μ Þ2 pffiffiffiffiffiffi exp ; 2σ 2 rσ 2π

ð10Þ

where  is the total volume of crystals per unit volume, r is the crystal size, μ is the natural logarithm of the mean size, and σ is the natural Table 1 Interfacial energies (σ), molar volumes (νm), and values of Λ for common secondary minerals. Mineral

Formula

σ a [J m− 2]

νmb [m3 mol− 1]

Λc [nm]

Hematite Goethite Kaolinite Gibbsite Quartz Barite Calcite Gypsum

Fe3O3 FeOOH Al2Si2O5(OH)4 Al(OH)3 SiO2 BaSO4 CaCO3 CaSO4.2H2O

1.200 1.600 0.200 0.140–0.483 0.350 0.135 0.097 0.026

3.04 × 10− 5 2.09 × 10− 5 9.93 × 10− 5 3.22 × 10− 5 2.27 × 10− 5 5.20 × 10− 5 3.69 × 10− 5 7.95 × 10− 5

29.5 27.0 16.0 3.6–12.6 6.4 5.7 2.9 1.7

a b c

From values compiled by Stumm and Morgan (1996). Calculated from Lide (1996). Calculated from Eq. (9) with T = 298 K.

logarithm of the standard deviation, which is not defined for purely unimodal size distributions. In weathering systems, crystal size populations can evolve continuously over time; however, the discussion here will be restricted to quantifying the instantaneous bulk rates of mineral precipitation in a system with a given crystal size distribution. Assuming that the crystals are spherical, r can also be expressed as r =

4πnr r3 ; 3

ð11Þ

where nr is a function describing the number of crystals of radius r per unit volume. The specific surface area associated with these crystals is therefore 2

Ar = 4πnr r :

ð12Þ

Combining Eqs. (11) and (12) to eliminate nr yields Ar = 3r = r;

ð13Þ

and it follows from Eq. (10) that ! 3 −ðln r−μ Þ2 Ar = 2 pffiffiffiffiffiffi exp : 2σ 2 r σ 2π

ð14Þ

14

S. Emmanuel, J.J. Ague / Chemical Geology 282 (2011) 11–18

growth)

Fig. 2. Schematic illustration of the difference between a standard kinetic approach and a model incorporating interfacial energy effects during mineral precipitation. When standard kinetics hold, the crystal growth rate (∂ r/∂ t) is uniform and independent of size. By contrast, when interfacial energy effects become important, large crystals grow while small crystals dissolve, significantly reducing the bulk reaction rate. As a result the rate laws governing the bulk reaction rates in the two scenarios will differ significantly.

different values of bulk supersaturation (S/S0), mean crystal size, and interfacial energy (Λ). The effect of the degree of supersaturation on reaction rate is demonstrated in Fig. 3 for a mean crystal size of 1 μm. Although a linear plot of rate versus S/S0 − 1 suggests that the two models do not differ significantly, a log–log plot shows distinct trends. Despite similar behavior at high supersaturations (S/S0 N 2), the two models diverge at lower values, with a much more rapid reduction in the reaction rate occurring in the interfacial energy model. This result reflects the fact that at low supersaturations, larger crystals continue to grow while small crystals dissolve, a process that greatly reduces the bulk reaction rate. Perhaps unexpectedly, this effect can lead to a reversal in the direction of the reaction, with the net precipitation rate becoming negative below a value of around S/S0 = 1.05. Although the negative value implies that net dissolution occurs, it is important to recognize that large crystals may continue to grow under such conditions. We note here that the non-linear dependence on reaction rate at low levels of supersaturation reported in some studies (e.g., Nagy and Lasaga, 1992; Burch et al., 1993; Arvidson and Luttge, 2010) is often attributed to a non-linear functional form of f(ΔG). It has been proposed that the transition from a linear to a non-linear dependence – which occurs at a critical free energy – is due to the opening of etch pits which facilitate continuous step movement on the dissolving mineral surface (Arvidson and Luttge, 2010). It is possible however that in some cases such non-linear behavior could be due to interfacial energy effects. Strikingly, the low levels of supersaturation (S/S0 b 1.5) and small

Such distributions can readily be evaluated to yield cumulative distribution curves for r and Ar (Fig. 1). Moreover, the full rate expression given in Eq. (8) becomes ∞

Re = k∫

0

3 −ðln r−μ Þ2 pffiffiffiffiffiffi exp 2 2σ 2 r σ 2π

!

β   S exp −2νm γ =Rg Tr −1 dr: S0 ð15Þ

We stress here that although different crystal geometries will lead to changes in Eq. (15), the overall functional form will be retained. As we were unable to identify an analytical solution, Eq. (15) was solved numerically using a Matlab code for a defined crystal size distribution (with parameters , σ and μ), the interfacial energy parameter Λ , bulk supersaturation S/S0, and rate constant k. Due to limitations in evaluating higher order kinetics with the present numerical algorithm, throughout this paper only first order reactions are assumed (i.e., β = 1). While the interfacial energy model can be used to describe the way in which solute concentrations and crystal populations evolve (see Appendix A), we will restrict the discussion here to the evaluation of the impact of supersaturation and crystal size on the net rate of precipitation. 3. Results and discussion 3.1. Assessing the impact of interfacial energy on reaction rates Two scenarios for mineral precipitation are considered in this study. In the first case, the rate is determined by the standard rate model (Eq. (4)), while in the second, mineral precipitation is influenced by interfacial energy effects (Eq. (15)). In the calculations, precipitation is considered to be fully reversible, such that crystals will either dissolve or grow depending on their effective state of supersaturation. The net rates of mineral precipitation for the two scenarios were evaluated for the

Fig. 3. Precipitation rate as a function of supersaturation (S/S0) for (a) linear axes and (b) log–log axes. The solid line indicates the rate for the standard kinetic model (Eq. (4)), while the dashed line represents the interfacial energy model. A value of 12.6 nm is assumed for Λ (gibbsite in Table 1). Although not shown due to the logarithmic scale, the net precipitation rate at low supersaturations (log10(S/S0 − 1) b − 1.2) becomes negative. The reaction rate is arbitrarily normalized to the reaction rate coefficient k and possesses units of m− 1. The total crystal volume fraction, , is 0.01, the mean crystal radius is 1 μm, and σ = 1.

S. Emmanuel, J.J. Ague / Chemical Geology 282 (2011) 11–18

particle size (1–10 μm) reported for some gibbsite precipitation experiments (Nagy and Lasaga, 1992) indicate that this may be feasible. Further differences between the two models can be seen by examining a plot of reaction rates as a function of both supersaturation and mean crystal size (Fig. 4). In the case of standard precipitation kinetics (Fig. 4a), lower levels of supersaturation reduce the net precipitation rate, while greater mean crystal size produces lower reaction rates due to reduced reactive surface areas. By contrast, a more complex pattern emerges when the effect of interfacial energy is taken into effect. When a value of Λ = 12.6 nm is adopted (gibbsite in Table 1), the interfacial energy model (Fig. 4b) mimics the standard kinetic model for high levels of supersaturation and large mean crystal sizes, while a shift from net precipitation to net dissolution occurs as supersaturation and mean crystal size decrease; importantly, this shift is extremely sharp, suggesting that relatively minor changes in supersaturation or crystal size distribution can result in a transition from a precipitation-dominated system to a dissolution-dominated one, or vice versa. Moreover, it is worth noting that although a low

a

log10R

2

log10(S/S0−1)

Λ = 0 nm 0

6 4 2

−2

0 −4 −6 −8

−2 −7

−6

−5

−4

−3

mean crystal radius [log10m]

log10(S/S0−1)

b

log10|Re|

2 0 −2

Λ = 12.6 nm

precipitation

6 4 2

dissolution

−2 −7

−6

−5

−4

log10|Re|

2 0 −2

6

Λ = 29.5 nm

precipitation

2

dissolution

0 −2 −7

−6

1 2νm γ : lnðS = S0 Þ Rg T

ð16Þ

Thus, it follows that the total surface area participating in the reactions (Areactive) is defined as ∞

Areactive = ∫ Ar dr: rcrit

ð17Þ

4

−4 −6 −8

rcrit =

−3

mean crystal radius [log10m]

log10(S/S0−1)

level of supersaturation favors dissolution, net dissolution can still occur in systems with S/S0 ∼ 10 when the mean crystal size is of the order of nanometers. Overall similar patterns are observed for higher values of Λ (Fig. 4c). We emphasize here that a system can still be dynamic, even in the absence of net precipitation or dissolution, as the dissolution of small crystals and the growth of larger ones will always cause the crystal size distribution to evolve. While it is well established that the growth of large crystals at the expense of smaller crystals is driven by interfacial energy, it is perhaps less well recognized that such a process can be important in supersaturated systems. Another important feature of the interfacial energy model is the near vertical contours at low S/S0, which indicate a decoupling of the reaction rate from the level of saturation. This non-intuitive behavior can be explained by inspection of Eq. (15): as S/S0 → 1, the term S S0 expð−2νm γ = RTr Þ≃expð−2νm γ = RTr Þ, and the rate equation thus becomes dominated by crystal size effects rather than the degree of saturation. To assess the conditions for which the reaction rate is not significantly affected by interfacial energy, it is instructive to examine the ratio of reaction rate calculated using the interfacial energy model to that calculated using standard kinetics (Re/R) as a function of mean crystal size and supersaturation (Fig. 5); when the interfacial energy model produces similar results to the standard rate equation, Re/R approaches unity; as the two equations diverge, however, the ratio gets smaller and eventually becomes negative. Inspection of Fig. 5 indicates that interfacial energy effects dominate for a large range of crystal sizes and level of supersaturation; by contrast, the standard kinetic model appears to be applicable for a region in which mean crystal size is greater than 1 μm and S/S0 N 1.01. Although the formulation of the interfacial energy model may seem complex, ultimately, the main reason for reduced precipitation rates is that the bulk of the surface area is associated with tiny crystals that are either not growing at all or are actually dissolving. Analogous to the critical nucleus size defined for homogeneous nucleation (Nielsen, 1964; Steefel and Van Cappellen, 1990), Eq. (6) can be used to define a critical threshold radius, rcrit, below which crystals no longer grow:

0 −4 −6 −8

c

15

−5

−4

−3

mean crystal radius [log10m] Fig. 4. Contour plot of reaction rate (R in the standard rate equation and Re in the interfacial energy model) as a function of supersaturation (S/S0) and mean crystal radius. (a) In the standard kinetic model (also equivalent to Λ = 0 nm in the interfacial energy model) only precipitation occurs; (b) in the interfacial energy model (Λ = 12.6 nm; gibbsite in Table 1) the phase space is separated into a region in which net precipitation takes place and another region in which dissolution dominates; the boundary between the two regions is indicated by the solid white line; and (c) interfacial energy model with Λ = 29.5 nm (hematite in Table 1). In all cases, the reaction rate is arbitrarily normalized to the reaction rate coefficient k and possesses units of m− 1. The total crystal volume fraction, , is 0.01 and σ = 1. In (b) and (c), absolute values are used to facilitate the use of a log scale.

From inspection, it can be seen that substitution of Areactive into Eq. (4) does not produce an equation that is equivalent to Eq. (7). Clearly, however, Areactive is an important parameter in determining the reaction rate, and can be used as a rough indicator of the magnitude of the interfacial effect. 3.2. Relationship between field rates and laboratory kinetics Secondary minerals produced during the weathering of primary minerals are commonly found to be submicron in scale. Thus if fluids in weathering systems possess low levels of supersaturation with respect to the secondary phases, we should expect that interfacial energy effects will significantly reduce the rates of mineral precipitation relative to rates predicted by standard rate equations in weathering systems. We can understand how this may help resolve the discrepancy between field and laboratory rates by comparing the conditions under which mineral precipitation is often explored in laboratory experiments with the conditions encountered in weathering systems. As shown in Fig. 6,

16

S. Emmanuel, J.J. Ague / Chemical Geology 282 (2011) 11–18

Fig. 5. Reaction rate ratio – defined as the rate evaluated using the interfacial energy model divided by the rate calculated with the standard rate equation (Re/R) – as a function of mean crystal radius and supersaturation. When the ratio is close to unity, interfacial effects do not significantly affect the standard rate equation; as the ratio approaches zero, interfacial effects reduce the effective rate by orders of magnitude, and when Re/R b 0, net dissolution occurs. As Re/R reaches very high negative values at low supersaturation and low mean crystal radius, the color range is restricted to − 1 b Re/R b 0.9. A value of 12.6 nm, corresponding to gibbsite in Table 1, is assumed for Λ .

much of the proposed “weathering zone” is not adequately described by standard kinetic formulations, while laboratory experiments fall primarily in the region described by standard reaction kinetics.

Fig. 6. Summary of system behavior as a function of mean crystal size and supersaturation. The domain is split into three regions based on Fig. 5: the region bounded by the blue line indicates the conditions for which the standard model serves as a reasonable approximation of the reaction kinetics (to within a factor of 2) marked as “standard precipitation kinetics”; the region lying between the blue line and the red line, indicates the zone in which precipitation dominates but the rate is significantly lower than that predicted by the standard kinetic model (marked as “reduced precipitation rate”); the region below the red line indicates dissolution dominated systems (marked as “dissolution dominated”). The orange zone represents the possible range of conditions for secondary mineral precipitation during weathering, while the light blue zone indicates the range of conditions typically encountered during weathering experiments. Note that while most of the “experimental” zone falls within the region of “standard precipitation kinetics”, much of the weathering zone falls outside.

Subsequently, interfacial energy effects should produce “anomalously” low precipitation rates for secondary minerals in weathering systems; moreover, as the kinetics of primary mineral dissolution are dependent on the rate of secondary mineral precipitation (Alekseyev et al., 1997; Lasaga, 1998; Ganor et al., 2007; Zhu and Lu, 2009), primary mineral weathering rates should also be much lower than those predicted by standard formulations. The overall mechanism is summarized schematically in Fig. 7. To further illustrate the potential impact of interfacial energy, we consider a dissolution experiment of a primary mineral in the laboratory during which secondary minerals are allowed to precipitate. Following an initial high level of supersaturation, a brief period of nucleation and rapid growth occurs during which the fluid saturation level drops quickly. Importantly though, the degree of supersaturation maintained artificially during the experiment is sufficiently high to ensure continued precipitation on all the secondary mineral crystal surfaces. In this case, Eq. (4) would serve as an adequate approximation of the rate law, and the primary mineral dissolution rate should be rapid. By contrast, in a naturally weathered system, nucleation might be followed by a period in which the degree of supersaturation of the secondary mineral is much lower than that in the laboratory experiment; as a result continued growth of small secondary mineral crystals will be stunted and the precipitation rate should proceed according to Eq. (8). As a result the rate of primary mineral dissolution will be significantly reduced, producing a discrepancy between field rates and the laboratory measurements. While a number of mechanisms have been invoked to account for the difference between field rates and laboratory experiments, perhaps the most often cited are: (i) low reactive surface areas of reacting minerals (Nugent et al., 1998; White and Brantley, 2003); (ii) slow reaction kinetics in near equilibrium systems (Burch et al., 1993; Oelkers et al., 1994; White and Brantley, 2003); and (iii) the slow rate of secondary mineral precipitation (Ganor et al., 2007; Zhu and Lu, 2009). The model presented here integrates aspects of all these explanations in a mechanistic way, offering a systematic framework for relating rates obtained in laboratory experiments to those obtained from field studies using analyses of crystal size distributions and saturation state data.

S. Emmanuel, J.J. Ague / Chemical Geology 282 (2011) 11–18

17

mineral dissolution

0

0

Fig. 7. Schematic representation of primary mineral weathering and secondary phase formation. Low S/S0 is likely to occur during natural weathering of minerals, while high S/S0 often characterizes laboratory experiments.

Given the potential magnitude of the interfacial energy effect and the fact that dissolution of secondary phases dominates for a range of conditions, it is worth speculating on how secondary minerals manage to persist in soils at all. Although the model we present here does not explicitly determine how a weathering system will change over time, we propose that systems could evolve along the route shown in Fig. 8. After initial exposure to weathering and initiation of nucleation of secondary phases, the level of supersaturation will drop rapidly due to the removal of solute via precipitation. As crystal growth progresses and supersaturation continues to fall, the system may reach a point at which no net precipitation occurs (red line in Fig. 8). If the system crosses this line, net dissolution will occur and the level of supersaturation will rise, returning the system to the regime of precipitation. On the other hand if supersaturation increases too much, the reaction rate will increase, which will act to reduce solute concentrations in the system. As a result, weathering systems might evolve slowly along a trajectory determined by the boundary between precipitation and dissolution dominated regimes. 4. Concluding remarks In this paper, we demonstrate that mineral precipitation rates in systems with a large number of micron and nanometer scale crystals can be much slower than rates predicted by standard kinetic equations. This effect – related to the interfacial energy associated with tiny crystals – is predicted to be strongly dependent on the level of supersaturation, becoming significant in systems close to equilibrium. Our calculations indicate that when such conditions prevail, standard reaction kinetics can easily produce reaction rates that are several orders of magnitude greater than those predicted by the interfacial energy model; moreover, at very low levels of supersaturation mineral precipitation may be superseded by net dissolution. The mechanism is expected to be important in geological systems, which are often near equilibrium, in contrast to laboratory rate experiments, which are typically conducted under far from equilibrium conditions. As dissolution rates of primary

Fig. 8. Possible crystallization path of secondary mineral phases in weathering systems. Following the initial nucleation of secondary phases, the level of supersaturation drops. As crystal growth progresses, supersaturation continues to fall. However, if the system crosses the red line, net dissolution will cause the level of supersaturation to rise; conversely, high supersaturation will increase the precipitation rate, reducing solute concentrations. To maintain stability, systems could evolve close to the line of zero net precipitation.

minerals are dependent on the precipitation rates of the secondary phases, it is proposed that this mechanism could account for the apparent discrepancy between the slow weathering rates observed under field conditions and those found in laboratory experiments. Clearly, the kinetic model presented here is a highly simplified representation of reaction rates in real systems, and a number of limitations associated with our simulations can be identified. The complex role of heterogeneous nucleation is not considered, while crystals have more complicated morphologies than the spherical geometry adopted here. Moreover, interfacial energy may vary significantly due to surface heterogeneities and differences between crystal faces. Crystal size distributions, too, are likely to be more complex than the log normal distribution adopted in the calculations; furthermore, such distributions will evolve with time so that the calculations presented here reflect a “snapshot” of instantaneous reaction rates; time integrated rates, calculated for systems in which mineral precipitation and dissolution are fully coupled, should further help elucidate the link between field and lab rates. We stress here that although the kinetic model presented in the paper has yet to be fully tested, it is based on a well founded physical mechanism. Moreover, the model offers an explanation for the difference between field weathering rates and laboratory kinetics which synthesizes two important common observations: weathering products are often tiny and many geological systems are very close to equilibrium. At present, detailed laboratory experiments are required to test the model and the impact of small secondary crystals on mineral dissolution rates.

Acknowledgments Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for support of this research. We thank two anonymous referees for their constructive comments.

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Appendix A. Evolving crystal size populations The equations on which reaction rates are based in Section 2 can also dictate the evolution of crystal size populations. Using a similar continuity equation to that described by Steefel and Van Cappellen (1990) and assuming spherical crystals, the rate of change of crystals of size r per unit volume can given by ∂nr ∂ðvnr Þ =− ð18Þ ∂t ∂r  β   S exp −2νm γ=Rg Tr −1 . However, this expression where v = kνm S0 differs from that adopted by Steefel and Van Cappellen (1990) (Eq. (19) in that paper) in that the definition of v here is dependent on crystal size and is therefore included in the derivative on the right hand side. Importantly, this means that even when S = S0, the crystal size population will continue to evolve and no additional terms are required to account for Ostwald ripening. This can be seen by expanding the right hand side to yield   ∂nr ∂n ∂v = − v r + nr ; ∂t ∂r ∂r

ð19Þ

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